Problem: Evaluate the definite integral. $\int^2_{4}\big(6x^2-2x+1\big)\,dx = $
First, use the power rule: $\int^2_{4}\big(6x^2-2x+1\big)\,dx~=~2x^3-x^2+x\Bigg|^2_{4}$ Second, plug in the limits of integration: $[2\cdot2^3-2^2+2]-[2\cdot4^3-4^2+4] = 14-116 = -102$. The answer: $\int^2_{4}\big(6x^2-2x+1\big)\,dx = -102$